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The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In...

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi (2021)

Czechoslovak Mathematical Journal

Let $𝔽_{q} [t]$ denote the polynomial ring over $𝔽_{q}$ , the finite field of $q$ elements. Suppose the characteristic of $𝔽_{q}$ is not $2$ or $3$ . We prove that there exist infinitely many $N \in ℕ$ such that the set ${f \in 𝔽_{q} [t] : deg f < N}$ contains a Sidon set which is an additive basis of order $3$ .

Simultane diophantische Approximationen abhängiger Größen in mehreren Metriken

F. Želudevič (1986)

Acta Arithmetica

Simultaneous Diophantine approximation and asymptotic formulae on manifolds

Glyn Harman (2003)

Acta Arithmetica

Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations

Ella I. Kovalevskaya, Vasily Bernik (2006)

Acta Mathematica Universitatis Ostraviensis

We prove an analogue of the convergence part of Khintchine’s theorem for the simultaneous inhomogeneous Diophantine approximation on the Veronese curve $(x, x^{2}, ..., x^{n})$ with respect to the different valuations. It is an extension of the author’s earlier results.

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Displaying 1 – 20 of 111

Schmidt's conjecture on normality for commuting matrices.

Semigroup-valued multiplicative functions

Sequences and dynamical systems associated with canonical approximation by rationals

Sequences with small discrepancy relative to n events

Sets of block structure and discrepancy estimates

S-expansions in dimension two

Shifted values of the largest prime factor function and its average value in short intervals

Sidon basis in polynomial rings over finite fields

Simultane diophantische Approximationen abhängiger Größen in mehreren Metriken

Simultaneous Diophantine approximation and asymptotic formulae on manifolds

Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations

Singularité des produits de Anzai associés aux fonctions caractéristiques d'un intervalle

Small digitwise perturbations of a number make it normal to unrelated bases.

Soma Estimates of the H...-Uniform Distribution.

Some applications of Elliot's mean-value theorem.

Some asymptotic results associated with a generalized Gauss-Kuzmin-type operator.

Some asymptotic results on extended sequences connected with the regular continued fraction.

Some Distribution Properties of 0,1-Sequences.

Some doubly exponential sums over ℤₘ

Some explicit constructions of sets with more sums than differences

Currently displaying 1 – 20 of 111